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Theorem ccatswrd 12663
Description: Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)
Assertion
Ref Expression
ccatswrd  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  =  ( S substr  <. X ,  Z >. ) )

Proof of Theorem ccatswrd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 swrdcl 12628 . . . . . 6  |-  ( S  e. Word  A  ->  ( S substr  <. X ,  Y >. )  e. Word  A )
21adantr 465 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. X ,  Y >. )  e. Word  A
)
3 swrdcl 12628 . . . . . 6  |-  ( S  e. Word  A  ->  ( S substr  <. Y ,  Z >. )  e. Word  A )
43adantr 465 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. Y ,  Z >. )  e. Word  A
)
5 ccatcl 12575 . . . . 5  |-  ( ( ( S substr  <. X ,  Y >. )  e. Word  A  /\  ( S substr  <. Y ,  Z >. )  e. Word  A
)  ->  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  e. Word  A )
62, 4, 5syl2anc 661 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  e. Word  A )
7 wrdf 12535 . . . 4  |-  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  e. Word  A  -> 
( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) : ( 0..^ ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) ) --> A )
8 ffn 5721 . . . 4  |-  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) : ( 0..^ ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) ) --> A  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( # `  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) ) )
96, 7, 83syl 20 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( # `  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) ) )
10 ccatlen 12576 . . . . . . 7  |-  ( ( ( S substr  <. X ,  Y >. )  e. Word  A  /\  ( S substr  <. Y ,  Z >. )  e. Word  A
)  ->  ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) )  =  ( ( # `  ( S substr  <. X ,  Y >. ) )  +  (
# `  ( S substr  <. Y ,  Z >. ) ) ) )
112, 4, 10syl2anc 661 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) )  =  ( ( # `  ( S substr  <. X ,  Y >. ) )  +  (
# `  ( S substr  <. Y ,  Z >. ) ) ) )
12 simpl 457 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  S  e. Word  A
)
13 simpr1 1003 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ( 0 ... Y ) )
14 simpr2 1004 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ( 0 ... Z ) )
15 simpr3 1005 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Z  e.  ( 0 ... ( # `  S ) ) )
16 fzass4 11732 . . . . . . . . . . . 12  |-  ( ( Y  e.  ( 0 ... ( # `  S
) )  /\  Z  e.  ( Y ... ( # `
 S ) ) )  <->  ( Y  e.  ( 0 ... Z
)  /\  Z  e.  ( 0 ... ( # `
 S ) ) ) )
1716biimpri 206 . . . . . . . . . . 11  |-  ( ( Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  -> 
( Y  e.  ( 0 ... ( # `  S ) )  /\  Z  e.  ( Y ... ( # `  S
) ) ) )
1817simpld 459 . . . . . . . . . 10  |-  ( ( Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  ->  Y  e.  ( 0 ... ( # `  S
) ) )
1914, 15, 18syl2anc 661 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ( 0 ... ( # `  S ) ) )
20 swrdlen 12632 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. X ,  Y >. ) )  =  ( Y  -  X ) )
2112, 13, 19, 20syl3anc 1229 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  ( S substr  <. X ,  Y >. ) )  =  ( Y  -  X ) )
22 swrdlen 12632 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. Y ,  Z >. ) )  =  ( Z  -  Y ) )
2312, 14, 15, 22syl3anc 1229 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  ( S substr  <. Y ,  Z >. ) )  =  ( Z  -  Y ) )
2421, 23oveq12d 6299 . . . . . . 7  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( # `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) )  =  ( ( Y  -  X )  +  ( Z  -  Y ) ) )
25 elfzelz 11699 . . . . . . . . . 10  |-  ( Y  e.  ( 0 ... Z )  ->  Y  e.  ZZ )
2614, 25syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ZZ )
2726zcnd 10977 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  CC )
28 elfzelz 11699 . . . . . . . . . 10  |-  ( X  e.  ( 0 ... Y )  ->  X  e.  ZZ )
2913, 28syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ZZ )
3029zcnd 10977 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  CC )
31 elfzelz 11699 . . . . . . . . . 10  |-  ( Z  e.  ( 0 ... ( # `  S
) )  ->  Z  e.  ZZ )
3215, 31syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Z  e.  ZZ )
3332zcnd 10977 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Z  e.  CC )
3427, 30, 33npncan3d 9972 . . . . . . 7  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( Y  -  X )  +  ( Z  -  Y
) )  =  ( Z  -  X ) )
3524, 34eqtrd 2484 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( # `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) )  =  ( Z  -  X
) )
3611, 35eqtrd 2484 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) )  =  ( Z  -  X ) )
3736oveq2d 6297 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( 0..^ (
# `  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) )  =  ( 0..^ ( Z  -  X
) ) )
3837fneq2d 5662 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( # `  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) )  <->  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( Z  -  X ) ) ) )
399, 38mpbid 210 . 2  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( Z  -  X ) ) )
40 swrdcl 12628 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. X ,  Z >. )  e. Word  A )
4140adantr 465 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. X ,  Z >. )  e. Word  A
)
42 wrdf 12535 . . . 4  |-  ( ( S substr  <. X ,  Z >. )  e. Word  A  -> 
( S substr  <. X ,  Z >. ) : ( 0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) --> A )
43 ffn 5721 . . . 4  |-  ( ( S substr  <. X ,  Z >. ) : ( 0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) --> A  ->  ( S substr  <. X ,  Z >. )  Fn  (
0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) )
4441, 42, 433syl 20 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. X ,  Z >. )  Fn  (
0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) )
45 fzass4 11732 . . . . . . . . 9  |-  ( ( X  e.  ( 0 ... Z )  /\  Y  e.  ( X ... Z ) )  <->  ( X  e.  ( 0 ... Y
)  /\  Y  e.  ( 0 ... Z
) ) )
4645biimpri 206 . . . . . . . 8  |-  ( ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z ) )  ->  ( X  e.  ( 0 ... Z
)  /\  Y  e.  ( X ... Z ) ) )
4746simpld 459 . . . . . . 7  |-  ( ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z ) )  ->  X  e.  ( 0 ... Z ) )
4813, 14, 47syl2anc 661 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ( 0 ... Z ) )
49 swrdlen 12632 . . . . . 6  |-  ( ( S  e. Word  A  /\  X  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. X ,  Z >. ) )  =  ( Z  -  X ) )
5012, 48, 15, 49syl3anc 1229 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  ( S substr  <. X ,  Z >. ) )  =  ( Z  -  X ) )
5150oveq2d 6297 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( 0..^ (
# `  ( S substr  <. X ,  Z >. ) ) )  =  ( 0..^ ( Z  -  X ) ) )
5251fneq2d 5662 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Z >. )  Fn  ( 0..^ (
# `  ( S substr  <. X ,  Z >. ) ) )  <->  ( S substr  <. X ,  Z >. )  Fn  ( 0..^ ( Z  -  X ) ) ) )
5344, 52mpbid 210 . 2  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. X ,  Z >. )  Fn  (
0..^ ( Z  -  X ) ) )
54 simpr 461 . . . . 5  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  x  e.  ( 0..^ ( Z  -  X ) ) )
5526, 29zsubcld 10981 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Y  -  X )  e.  ZZ )
5655adantr 465 . . . . 5  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( Y  -  X )  e.  ZZ )
57 fzospliti 11839 . . . . 5  |-  ( ( x  e.  ( 0..^ ( Z  -  X
) )  /\  ( Y  -  X )  e.  ZZ )  ->  (
x  e.  ( 0..^ ( Y  -  X
) )  \/  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) ) )
5854, 56, 57syl2anc 661 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( x  e.  ( 0..^ ( Y  -  X ) )  \/  x  e.  ( ( Y  -  X
)..^ ( Z  -  X ) ) ) )
592adantr 465 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( S substr  <. X ,  Y >. )  e. Word  A )
604adantr 465 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( S substr  <. Y ,  Z >. )  e. Word  A )
6121oveq2d 6297 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( 0..^ (
# `  ( S substr  <. X ,  Y >. ) ) )  =  ( 0..^ ( Y  -  X ) ) )
6261eleq2d 2513 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  e.  ( 0..^ ( # `  ( S substr  <. X ,  Y >. ) ) )  <-> 
x  e.  ( 0..^ ( Y  -  X
) ) ) )
6362biimpar 485 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  x  e.  ( 0..^ ( # `  ( S substr  <. X ,  Y >. ) ) ) )
64 ccatval1 12577 . . . . . . 7  |-  ( ( ( S substr  <. X ,  Y >. )  e. Word  A  /\  ( S substr  <. Y ,  Z >. )  e. Word  A  /\  x  e.  (
0..^ ( # `  ( S substr  <. X ,  Y >. ) ) ) )  ->  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. X ,  Y >. ) `  x
) )
6559, 60, 63, 64syl3anc 1229 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. X ,  Y >. ) `
 x ) )
66 simpll 753 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  S  e. Word  A )
67 simplr1 1039 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  X  e.  ( 0 ... Y
) )
6819adantr 465 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  Y  e.  ( 0 ... ( # `
 S ) ) )
69 simpr 461 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  x  e.  ( 0..^ ( Y  -  X ) ) )
70 swrdfv 12633 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) )  /\  x  e.  ( 0..^ ( Y  -  X
) ) )  -> 
( ( S substr  <. X ,  Y >. ) `  x
)  =  ( S `
 ( x  +  X ) ) )
7166, 67, 68, 69, 70syl31anc 1232 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( ( S substr  <. X ,  Y >. ) `  x )  =  ( S `  ( x  +  X
) ) )
7265, 71eqtrd 2484 . . . . 5  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( S `  ( x  +  X
) ) )
732adantr 465 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( S substr  <. X ,  Y >. )  e. Word  A
)
744adantr 465 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( S substr  <. Y ,  Z >. )  e. Word  A
)
7521, 35oveq12d 6299 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( # `  ( S substr  <. X ,  Y >. ) )..^ ( ( # `  ( S substr  <. X ,  Y >. ) )  +  (
# `  ( S substr  <. Y ,  Z >. ) ) ) )  =  ( ( Y  -  X )..^ ( Z  -  X ) ) )
7675eleq2d 2513 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  e.  ( ( # `  ( S substr  <. X ,  Y >. ) )..^ ( (
# `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) ) )  <->  x  e.  (
( Y  -  X
)..^ ( Z  -  X ) ) ) )
7776biimpar 485 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  x  e.  ( ( # `  ( S substr  <. X ,  Y >. ) )..^ ( (
# `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) ) ) )
78 ccatval2 12578 . . . . . . 7  |-  ( ( ( S substr  <. X ,  Y >. )  e. Word  A  /\  ( S substr  <. Y ,  Z >. )  e. Word  A  /\  x  e.  (
( # `  ( S substr  <. X ,  Y >. ) )..^ ( ( # `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. Y ,  Z >. ) `
 ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) ) ) )
7973, 74, 77, 78syl3anc 1229 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. Y ,  Z >. ) `  (
x  -  ( # `  ( S substr  <. X ,  Y >. ) ) ) ) )
80 simpll 753 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  S  e. Word  A
)
81 simplr2 1040 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  Y  e.  ( 0 ... Z ) )
82 simplr3 1041 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  Z  e.  ( 0 ... ( # `  S ) ) )
8321oveq2d 6297 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  =  ( x  -  ( Y  -  X ) ) )
8483adantr 465 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  =  ( x  -  ( Y  -  X ) ) )
8534oveq2d 6297 . . . . . . . . . . 11  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( Y  -  X )..^ ( ( Y  -  X
)  +  ( Z  -  Y ) ) )  =  ( ( Y  -  X )..^ ( Z  -  X
) ) )
8685eleq2d 2513 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  e.  ( ( Y  -  X )..^ ( ( Y  -  X )  +  ( Z  -  Y
) ) )  <->  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) ) )
8786biimpar 485 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  x  e.  ( ( Y  -  X
)..^ ( ( Y  -  X )  +  ( Z  -  Y
) ) ) )
8832, 26zsubcld 10981 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Z  -  Y )  e.  ZZ )
8988adantr 465 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( Z  -  Y )  e.  ZZ )
90 fzosubel3 11859 . . . . . . . . 9  |-  ( ( x  e.  ( ( Y  -  X )..^ ( ( Y  -  X )  +  ( Z  -  Y ) ) )  /\  ( Z  -  Y )  e.  ZZ )  ->  (
x  -  ( Y  -  X ) )  e.  ( 0..^ ( Z  -  Y ) ) )
9187, 89, 90syl2anc 661 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( x  -  ( Y  -  X
) )  e.  ( 0..^ ( Z  -  Y ) ) )
9284, 91eqeltrd 2531 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  e.  ( 0..^ ( Z  -  Y ) ) )
93 swrdfv 12633 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  Y  e.  (
0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  /\  ( x  -  ( # `
 ( S substr  <. X ,  Y >. ) ) )  e.  ( 0..^ ( Z  -  Y ) ) )  ->  (
( S substr  <. Y ,  Z >. ) `  (
x  -  ( # `  ( S substr  <. X ,  Y >. ) ) ) )  =  ( S `
 ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y ) ) )
9480, 81, 82, 92, 93syl31anc 1232 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( S substr  <. Y ,  Z >. ) `
 ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) ) )  =  ( S `  (
( x  -  ( # `
 ( S substr  <. X ,  Y >. ) ) )  +  Y ) ) )
9583oveq1d 6296 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y )  =  ( ( x  -  ( Y  -  X )
)  +  Y ) )
9695adantr 465 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y )  =  ( ( x  -  ( Y  -  X )
)  +  Y ) )
97 elfzoelz 11811 . . . . . . . . . . 11  |-  ( x  e.  ( ( Y  -  X )..^ ( Z  -  X ) )  ->  x  e.  ZZ )
9897zcnd 10977 . . . . . . . . . 10  |-  ( x  e.  ( ( Y  -  X )..^ ( Z  -  X ) )  ->  x  e.  CC )
9998adantl 466 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  x  e.  CC )
10027, 30subcld 9936 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Y  -  X )  e.  CC )
101100adantr 465 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( Y  -  X )  e.  CC )
10227adantr 465 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  Y  e.  CC )
10399, 101, 102subadd23d 9958 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( x  -  ( Y  -  X ) )  +  Y )  =  ( x  +  ( Y  -  ( Y  -  X ) ) ) )
10427, 30nncand 9941 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Y  -  ( Y  -  X
) )  =  X )
105104oveq2d 6297 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  +  ( Y  -  ( Y  -  X )
) )  =  ( x  +  X ) )
106105adantr 465 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( x  +  ( Y  -  ( Y  -  X )
) )  =  ( x  +  X ) )
10796, 103, 1063eqtrd 2488 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y )  =  ( x  +  X ) )
108107fveq2d 5860 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( S `  ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y
) )  =  ( S `  ( x  +  X ) ) )
10979, 94, 1083eqtrd 2488 . . . . 5  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( S `  ( x  +  X ) ) )
11072, 109jaodan 785 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  ( x  e.  ( 0..^ ( Y  -  X ) )  \/  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( S `  ( x  +  X
) ) )
11158, 110syldan 470 . . 3  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( S `  ( x  +  X
) ) )
112 simpll 753 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  S  e. Word  A )
11348adantr 465 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  X  e.  ( 0 ... Z
) )
114 simplr3 1041 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  Z  e.  ( 0 ... ( # `
 S ) ) )
115 swrdfv 12633 . . . 4  |-  ( ( ( S  e. Word  A  /\  X  e.  (
0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  /\  x  e.  ( 0..^ ( Z  -  X
) ) )  -> 
( ( S substr  <. X ,  Z >. ) `  x
)  =  ( S `
 ( x  +  X ) ) )
116112, 113, 114, 54, 115syl31anc 1232 . . 3  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( ( S substr  <. X ,  Z >. ) `  x )  =  ( S `  ( x  +  X
) ) )
117111, 116eqtr4d 2487 . 2  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. X ,  Z >. ) `
 x ) )
11839, 53, 117eqfnfvd 5969 1  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  =  ( S substr  <. X ,  Z >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   <.cop 4020    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495    + caddc 9498    - cmin 9810   ZZcz 10871   ...cfz 11683  ..^cfzo 11806   #chash 12387  Word cword 12516   concat cconcat 12518   substr csubstr 12520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-hash 12388  df-word 12524  df-concat 12526  df-substr 12528
This theorem is referenced by:  wrdcctswrd  12672  swrdccatwrd  12675  wrdeqcats1  12681  wrdeqs1cat  12682  splid  12711  splval2  12715  swrds2  12865  efgredleme  16740  efgredlemc  16742  efgcpbllemb  16752  frgpuplem  16769  wrdsplex  28473
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